Integration and resolution of differential equations in finite terms

Abstract

[EN] When we are introduced to integrals in high school we learn some methods to find primitives of functions. We may try to use these methods to integrate very simple functions such as e x 2 , e x/x or (sin x)/x failing once and again and wonder if we are doing something wrong and why we are unable to find the result. The reason is that this result “does not” exist, in the sense that there is not a “prettier” way to describe the function R e x 2 dx than R e x 2 dx itself. Certainly, it can be written as an infinite series, but it cannot be expressed as a combination of finitely many elementary functions (which we usually understand as radical, exponential and trigonometric functions and their inverses). Describing if an integral can be expressed in this way is what is known as the problem of integration in finite terms. Similarly, once we learn various methods to solve differential equations, we may try to use them to solve simple equations like Y 00 − xY = 0 with no success. Again the problem is that this equation “does not” have a solution in a similar sense as above. But now the impossibility is even stronger: the solution of the equation can not be given even if we allow also integral symbols in the final expression (notice that in this sense the previous problem would trivially have a solution). This problem is studied in differential Galois theory, an analogue of classical Galois theory, that studies solutions of homogenous linear differential equations instead of polynomial equations. In order to be able to make more precise this problem we need to formalize the concepts, which will allow us to work on an abstract setting.